Geometrical Meaning of Rank of 2x2 Matrix | What is Rank? (Linear Algebra) (Part 1)

Welcome friends, in this video we are going to understand the geometrical meaning of rank of 2 cross 2 square matrix. Working rule for finding rank of matrix the number of non-zero rows in the row echelon form of a matrix A is called the rank of a matrix. Example A is equal to 1 2 minus 3 minus 6. We are going to find the rank of matrix A. For that, we convert matrix A into its row echelon form and then calculate the number of non-zero in the row echelon form. So, apply the row operation R2 equal to R2 plus 3R1. That means, multiply first row by 3 and add to the corresponding entries of second row. So, plus 3 added to the minus 3, plus 6 added to the minus 6, then it will be 0. So this is row echelon form of matrix A and that row echelon form contains Only one non-zero therefore rank is one. Now we are going to understand the geometrical meaning of rank one into cross two matrix. For that we consider every row as a vector of R2 that means first row one two and second row minus three minus six is a vector of R2 and we are going to plot on plane R2. So first vector one two and the second minus three minus six Now, we can observe that both the position vector lie on the line which is a multiple of 1, 2. Therefore, we conclude that if the rank of 2 cross 2 matrix is 1, that means matrix made by two collinear position vectors of R2. Second example, A equal to 1 minus 3, 3, 4. Now, we convert into the row echelon form. So, row operation R2 equal to R2 minus 3 R1. Multiply first row by minus 3 and add to the second row to minus 3 plus 3, 0 plus 9 plus 4, 13. This is a row echelon form of matrix A and the number of non-zero in the row echelon form are two. Therefore, rank of A is equal to 2. Now, similarly, we consider first and second row as a vector of R2. Therefore, 1 minus 3 is a vector of R2 and 3, 4 is also a vector of R2. Now plotting this point. On R2, we have 1 minus 3 and the second 3, 4. Both the vectors lie in the different direction. Therefore, we cannot find any line which passing through both the position vector. So, we conclude that if the rank of 2 cross 2 matrix is 2, then matrix made by two non-collinear position vectors of R2. Now, summarize the result. Rank of 2 cross 2 matrix is 0. Then, Matrix should be 0 matrix. If rank of matrix is 1, then matrix made by 2 collinear position vectors. And if rank of matrix is 2, then matrix made by 2 non-collinear position vectors in R2. Thanks for watching. For more videos, like, subscribe, press the bell icon of NextGenMath.